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| موضوع: كتاب Statics and Mechanics of Materials السبت 07 يوليو 2012, 11:30 pm | |
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أخواني في الله أحضرت لكم كتاب Statics and Mechanics of Materials 5th Edition R. C. Hibbeler SI Conversion by Kai Beng Yap
و المحتوى كما يلي :
CONTENTS General Principles 21 Chapter Objectives 21 1.1 Mechanics 21 1.2 Fundamental Concepts 22 1.3 The International System of Units 26 1.4 Numerical Calculations 28 1.5 General Procedure for Analysis 29 1 Force Vectors 35 Chapter Objectives 35 2.1 Scalars and Vectors 35 2.2 Vector Operations 36 2.3 Vector Addition of Forces 38 2.4 Addition of a System of Coplanar Forces 49 2.5 Cartesian Vectors 58 2.6 Addition of Cartesian Vectors 61 2.7 Position Vectors 70 2.8 Force Vector Directed Along a Line 73 2.9 Dot Product 81 2 Force System Resultants 97 Chapter Objectives 97 3.1 Moment of a Force—Scalar Formulation 97 3.2 Cross Product 101 3.3 Moment of a Force—Vector Formulation 104 3.4 Principle of Moments 108 3.5 Moment of a Force about a Specified Axis 119 3.6 Moment of a Couple 128 3.7 Simplification of a Force and Couple System 138 3.8 Further Simplification of a Force and Couple System 149 3.9 Reduction of a Simple Distributed Loading 161 3 Equilibrium of a Rigid Body 175 Chapter Objectives 175 4.1 Conditions for Rigid-Body Equilibrium 175 4.2 Free-Body Diagrams 177 4.3 Equations of Equilibrium 187 4.4 Two- and Three-Force Members 193 4.5 Free-Body Diagrams 203 4.6 Equations of Equilibrium 208 4.7 Characteristics of Dry Friction 218 4.8 Problems Involving Dry Friction 222 4 Structural Analysis 241 Chapter Objectives 241 5.1 Simple Trusses 241 5.2 The Method of Joints 244 5.3 Zero-Force Members 250 5.4 The Method of Sections 257 5.5 Frames and Machines 266 5 Center of Gravity, Centroid, and Moment of Inertia 287 Chapter Objectives 287 6.1 Center of Gravity and the Centroid of a Body 287 6.2 Composite Bodies 301 6.3 Moments of Inertia for Areas 310 6.4 Parallel-Axis Theorem for an Area 311 6.5 Moments of Inertia for Composite Areas 319 616 contents Torsion 471 Chapter Objectives 471 10.1 Torsional Deformation of a Circular Shaft 471 10.2 The Torsion Formula 474 10.3 Power Transmission 482 10.4 Angle of Twist 492 10.5 Statically Indeterminate Torque-Loaded Members 506 10 Bending 517 Chapter Objectives 517 11.1 Shear and Moment Diagrams 517 11.2 Graphical Method for Constructing Shear and Moment Diagrams 524 11.3 Bending Deformation of a Straight Member 543 11.4 The Flexure Formula 547 11.5 Unsymmetric Bending 562 11 Transverse Shear 577 Chapter Objectives 577 12.1 Shear in Straight Members 577 12.2 The Shear Formula 578 12.3 Shear Flow in Built-Up Members 596 12 Combined Loadings 609 Chapter Objectives 609 13.1 Thin-Walled Pressure Vessels 609 13.2 State of Stress Caused by Combined Loadings 616 13 Mechanical Properties of Materials 397 Chapter Objectives 397 8.1 The Tension and Compression Test 397 8.2 The Stress–Strain Diagram 399 8.3 Stress–Strain Behavior of Ductile and Brittle Materials 403 8.4 Strain Energy 407 8.5 Poisson’s Ratio 416 8.6 The Shear Stress–Strain Diagram 418 8 Axial Load 429 Chapter Objectives 429 9.1 Saint-Venant’s Principle 429 9.2 Elastic Deformation of an Axially Loaded Member 431 9.3 Principle of Superposition 446 9.4 Statically Indeterminate Axially Loaded Members 446 9.5 The Force Method of Analysis for Axially Loaded Members 453 9.6 Thermal Stress 459 9 Stress and Strain 329 Chapter Objectives 329 7.1 Introduction 329 7.2 Internal Resultant Loadings 330 7.3 Stress 344 7.4 Average Normal Stress in an Axially Loaded Bar 346 7.5 Average Shear Stress 353 7.6 Allowable Stress Design 364 7.7 Deformation 379 7.8 Strain 380 7contents 17 Stress and Strain Transformation 637 Chapter Objectives 637 14.1 Plane-Stress Transformation 637 14.2 General Equations of Plane-Stress Transformation 642 14.3 Principal Stresses and Maximum In-Plane Shear Stress 645 14.4 Mohr’s Circle—Plane Stress 661 14.5 Absolute Maximum Shear Stress 673 14.6 Plane Strain 679 14.7 General Equations of Plane-Strain Transformation 680 *14.8 Mohr’s Circle—Plane Strain 688 *14.9 Absolute Maximum Shear Strain 696 14.10 Strain Rosettes 698 14.11 Material Property Relationships 700 14 Design of Beams and Shafts 717 Chapter Objectives 717 15.1 Basis for Beam Design 717 15.2 Prismatic Beam Design 720 15 Deflection of Beams and Shafts 735 Chapter Objectives 735 16.1 The Elastic Curve 735 16.2 Slope and Displacement by Integration 739 *16.3 Discontinuity Functions 757 16.4 Method of Superposition 768 16.5 Statically Indeterminate Beams and Shafts—Method of Superposition 776 16 17 Buckling of Columns 795 Chapter Objectives 795 17.1 Critical Load 795 17.2 Ideal Column with Pin Supports 798 17.3 Columns Having Various Types of Supports 804 *17.4 The Secant Formula 816 Appendix A Mathematical Review and Expressions 828 B Geometric Properties of An Area and Volume 832 C Geometric Properties of Wide-Flange Sections 834 D Slopes and Deflections of Beams 837 Preliminary Problems Solutions 839 Fundamental Problems Solutions and Answers 858 Selected Answers 891 Index 916 INDEX A Absolute maximum shear strain, 696–697, 713 Absolute maximum shear stress, 673–676, 712 Allowable stress (sallow), 364–365, 723, 733 Allowable stress design (ASD), 364–371, 393 Angle of twist, 472–473, 492–499, 513 circular shafts, 472–473 constant cross-sectional area (A), 493–494 internal torque and, 492–493, 496 multiple torques, 494 procedure for analysis of, 496 sign convention for, 495 torsion and, 472–473, 492–499, 513 torsional deformation and, 472–473 Angles (u), 35, 59–61, 82, 92–93, 219–221. See also Angle of twist Cartesian vectors, 58–61 coordinate direction (a, b, g), 59–60, 92–93 dot products and, 82, 93 dry friction (f), 219–220 horizontal, 60–61 kinetic friction (fk), 220–221 line of action (direction) and, 35 static friction (fs), 219, 221 vertical (f), 60–61 Anisotropic material, 346 Area (A), 290, 310–314, 319–321, 325–326, 403, 432–433, 467, 832–833 centroid (C) of, 290, 325, 832–833 composite, 319–321, 326 constant cross-sectional, 432–433, 467 moment of inertia (I ) for, 310–314, 319–321, 326, 832–833 parallel-axis theorem for, 311–314, 326 percent reduction in, 403 procedures for analysis of, 312, 319 Arrow notation, 35, 49, 91 Axes, 98, 119–123, 161–165, 170, 543–546, 548, 562–563 bending applied to, 543–546 direction and, 98 distributed loading along single, 161–165 line of action, 119, 170 longitudinal, 543–545 moment, 98 moment of force about, 119–123, 170 neutral, 543, 548, 562, 565 principal, 562–563 projection, 120 resultant force on, 119–123, 170 scalar analysis of, 119 unsymmetric bending and, 562–563 vector analysis of, 120–121 Axial loads, 346–352, 392, 428–469, 795–797, 816–821, 825 bars, 346–352, 392 buckling from, 795–797, 816–821, 825 columns, 795–797, 816–821, 825 constant cross-sectional area (A), 432–433, 467 cross section of, 346 displacement (d), 431–438, 447–452, 467 eccentric applications, 816–821, 827 elastic deformation in members, 431–438, 467 force (flexibility) method of analysis, 453–454 principle of superposition for, 446, 467 procedures for analysis of, 434, 448, 453–454 relative displacement (d), 431–438, 466 Saint-Venant’s principle for, 429–431, 466 sign convention for, 433, 467 statically indeterminate members, 446–452, 467 stress (s) and, 346–352, 392 thermal stress and, 459–462, 467 Axis of symmetry, 543, 562 B Ball-and-socket joints, 203 Base units, 26 Beams, 516–575, 576–607, 716–733, 734–793, 834–839 bearing plates, 718 bending, 516–575 built-up members, 596–600, 605, 722, 733 cantilevered, 839 concentrated forces and moments, 526 deflection of, 718, 734–793, 838–839 deformation of, 543–546 design of, 716–733 discontinuity functions for, 757–765, 790 distributed loads, 524–526 equations for, 838–839 fastener spacing, 597, 605 flexure formula for, 547–553, 573 force (flexibility) method for, 777–780 glulam, 722 graphical representations of, 524–533, 572 load-displacement relationships, 777–780 longitudinal shear stress in, 577–578 916Index 917 Beams (continued) method of integration for, 739–749, 790 method of superposition for, 768–772, 776–785, 791 moment diagrams, 735–749, 790 plate girders, 719 prismatic, 720–727 procedures for analysis of, 519, 527, 550, 584, 723, 742, 762, 780 section modulus (S), 720 shear and moment diagrams, 517–533, 572 shear flow (q), 596–600, 605 shear formula for, 578–589, 605 sign convention for, 518, 526, 572 simply supported, 838 statically indeterminate members, 776–785, 791 steel sections, 721 stress trajectories, 719 transverse shear, 576–607 unsymmetric bending, 562–568, 573 warping, 578 wide-flange sections, 834–837 wood sections, 721 Bearing plates, 718 Bearing supports, 203 Bending, 516–575 axis of symmetry, 543, 562 beams, 516–575 deformation, 543–546 flexure formula for, 547–553, 573 neutral axis, 543, 548, 562, 565 neutral surface, 543, 562 principal axis, 562–563 procedures for analysis of, 519, 527, 550 shear and moment diagrams, 517–533, 572 straight members, 543–546, 573 unsymmetric, 562–568, 573 Bending moment (M), 331–332, 526, 543–546, 548–549, 562–568, 572–573 arbitrarily applied, 564 deformation of straight members, 543–546, 572 flexure formula for, 548–549, 573 internal resultant loadings, 331–332 principal axis, applied to, 562–563, 573 shear and moment diagram regions of concentration, 526 unsymmetric bending, 562–568, 573 Bending stress, 723, 733 Biaxial stress, 611 Bifurcation point, 797 Bridge trusses, 242 Brittle materials, 405, 424 Buckling, 794–827 axial loads, 795–797, 816–821, 825 bifurcation point, 797 columns, 794–827 critical load (Pcr) for, 795–797, 825 eccentric loading, 816–821, 825 equilibrium and, 796–797 lateral deflection, 795–797 Built-up members, 596–600, 605, 722–723, 733 design of, 722–723, 733 fastener spacing, 597, 605, 723 plate girders for, 722 shear flow (q) in, 596–600, 605 Bulk modulus (k), 703, 713 C Cartesian stress and strain components, 381 Cartesian vectors, 50, 58–64, 70–75, 81, 92–93, 102–103, 105–107, 169 addition of, 61 coordinate direction angles (a), 59–60 coplanar forces, notation of, 50 cross product, formulation of, 102–103, 169 direction of, 58–61, 92–93 dot product and, 81, 93 force vectors, 73–75 horizontal angle (u), 60–61 magnitude of, 59, 92 position vectors for, 70–72, 169 rectangular components, 58 resultant forces, formulation of, 61–62, 93 resultant moments, formulation of, 105–107, 169 right-hand rule for, 58, 101–102 three-dimensional components, 58–61 unit vectors, 58–61, 73–75, 92 vertical angle (f), 60–61 x, y, z coordinates, 59–60, 70 Center of gravity (G), 180, 287–289, 291, 301–302, 325 composite bodies, 301–302, 325 free-body diagram location, 180 procedures for analysis of, 291, 302 rigid-body equilibrium and, 180 specific weight (constant density) and, 301 weight (W) and, 180, 287–289, 325 Centroid (C), 162, 171, 289–294, 325, 832–833 area (A), 290, 325, 832–833 distributed loading, 162, 171 procedure for analysis of, 291 volume (V), 289, 325, 832–833918 Index Circular shafts, torsional deformation of, 471–473, 513 Coefficients of friction (μ), 219–221, 237 Cohesive material, 344 Collinear vectors, 37, 91 Columns, 794–827 critical load (Pcr) of, 795–797, 800–801, 804, 820, 825 deflection equations for, 798–801, 817–818 design of, 820 eccentric loading, 816–821, 825 effective length, 805 Euler load, 800, 825 fixed-supported (braced), 804 ideal, 798–803 lateral deflection of, 795–797 pin-supported, 798–803 Secant formula for, 816–821, 825 Combined loadings, 608–635 biaxial stress, 611 cylindrical (hoop) stress, 610–611, 632 cylindrical vessels, 610–611, 632 procedure for analysis of, 616–617 radial stress, 611 spherical vessels, 611, 632 state of stress caused by, 616–623, 632 thin-walled pressure vessels, 609–612, 632 Compatibility (kinematic) conditions, 447 Component vectors, 36, 38–39, 49–54 Composite bodies, 301–304, 319–321, 325–326 area (A), 319–321, 326 center of gravity (G) and centroid of, 301–302, 325 moment of inertia for, 319–321, 326 procedures for analysis of, 302, 319 specific weight and, 301 Compression test, 397–398 Compressive forces, 243–244, 257–258 Concentrated force, 22 Concurrent forces, 50, 61, 93, 149, 193 couple system simplification, 149 resultants, 50, 92 three-force member equilibrium, 193 Continuous material, 344 Coplanar forces, 49–54, 92, 138–140, 149, 161, 177–185, 187–194, 332 Cartesian vector notation, 50 couple system simplification, 138–140, 149 distributed loads, 161 equations of equilibrium for, 187–192 internal resultant loadings, 332 procedures for analysis of, 182, 188 resultants, 50–51, 92 rigid-body equilibrium and, 177–185, 187–194 scalar notation, 49 support reactions for, 177–185 three-force members, 193–194 two-force members, 193–194 Cosine law, 40 Coulomb friction, 218. See also Dry friction Couple, 128–133, 138–143, 149–154, 170–171, 177 concurrent force systems, 149 coplanar force systems, 138–140, 149, 177 equivalent, 129 moment of, 128–133, 170 parallel force systems, 150 procedures for analysis of, 140, 150 resultant moment (MR), 129–130 rigid-body equilibrium and, 177 scalar formulation, 128 system simplification, 138–143, 149–154, 171 vector formulation, 128, 170 Critical load (Pcr), 795–797, 800–801, 804, 820, 825 bifurcation point, 797 buckling and, 795–797, 825 column design and, 820 deflection and, 798–801 equilibrium and, 796–797 Euler load, 800, 825 fixed supported (braced), 804 pin-supported columns, 800–801, 825 Cross product, 101–103, 169 Cross sections, 330–332, 344–348, 392, 432–433, 467 axially loads, 346–348, 432–433, 467 constant area (A), 432–433, 467 internal resultant loading, 330–332, 392 stress distribution, 344–348 transverse shear moment (Q), 580–581 Cylindrical (hoop) stress, 610–611, 632 Cylindrical vessels, 610–611, 632 D Deflection, 718, 734–793, 795–797, 816–821, 825, 838–839. See also Buckling beams, 718, 734–793, 838–839 boundary conditions, 740 cantilevered beams, 838 column buckling, 795–797, 816–821, 825 continuity conditions, 740 coordinates for, 741 critical load (Pcr), 795–797, 825Index 919 Deflection (continued) discontinuity functions for, 757–765, 790 eccentric loading, 816–821, 825 elastic curve, 735–739, 790, 838–839 flexural rigidity (EI ), 740 lateral (buckling), 795–797 method of integration for, 739–749, 790 method of superposition for, 768–772, 776–785, 791 moment–curvature relationship, 738 moment diagrams, 735–749, 790 procedures for analysis of, 742, 762, 780 Secant formula for, 816–821, 825 sign conventions for, 751 simply supported beams, 838 slope equations, 739–749, 790, 838–839 statically indeterminate members, 776–785, 791 Deformation, 346, 379–385, 393, 397–398, 400–401, 403–417, 424–425, 429–438, 543–546, 573, 578 axially loaded members, 431–438 bending, 543–546, 573 brittle materials, 405, 424 circular shafts, 471–473, 513 displacement (d) and, 431–438, 447, 452, 467 ductile materials, 403–404, 424 elastic, 431–438, 467 localized, 429–430 necking, 401 permanent, 400, 424 plastic, 400 Poisson’s ratio (ν) for, 416–417, 425 relative displacement (d), 431–438 Saint-Venant’s principle for, 429–431, 466 small strain analysis, 382 straight members (beams), 543–546, 573 strain and, 380–385, 393 strain energy from, 407–411, 425 strain hardening, 401 stress–strain diagrams for, 400–401, 424 tension and compression tests for, 397–398 torsional, 471–473, 513 twisting, 471–473 uniform, 346 warping, 578 yielding, 400 Degree of indeterminacy, 776 Derivatives, 830 Design. See Structural design Determinant notation, 103 Dilatation (e), 702–703, 713 Dimensional homogeneity, 28 Dimensionless quantity, 380, 393 Direction, 35, 38, 49, 58–61, 70–71, 73, 91–93, 98, 101–102, 104, 128, 169–170, 244, 250, 258, 719 angle (u) for, 35, 60–61 arrow notation for, 49 Cartesian vectors, 58–61, 92–93 coordinate angles (a, b, g), 59–60, 92–93 cosines, 59–60 couple moments, 128 cross product, 101–102 force and, 73–75, 82, 92–93, 98, 102–103, 104, 169–170 by inspection, 244, 250, 258 line of action, 35 moment axis, 98, 170 position vectors, 70–71 resultant forces, 38–39 resultant moments (MR), 98, 102–103, 169–170 right-hand rule for, 58, 98, 102–103, 104, 128, 169 scalar formulation and, 98, 169 sense of, 35, 91 sign convention for, 98 stress trajectories, 719 truss member forces, 244, 250, 258 unit vectors, 50, 58, 73–75, 92 vectors and, 35, 38, 49, 58–61, 70–71, 73, 91–92, 102–103, 104 Discontinuity functions, 757–765, 790 applications of, 761 deflection and, 757–765, 790 Macaulay functions, 758 procedure for analysis of, 762 singularity functions, 759–760 Displacement (d), 431–438, 447–452, 467 axially loaded members, 431–438, 448–452 compatibility (kinematic) conditions for, 447–452, 467 constant cross-sectional area (A), 432–433, 467 principle of superposition for, 446, 467 procedure for analysis of, 434 relative, 431–438 statically indeterminate members, 447–452, 467 Distributed loading, 161–165, 171, 344–345, 477, 524–526, 757–765, 790 axially loaded members, 346–348 axis (single), along, 161–165 centroid (C) location, 162, 171 coplanar, 161–165, 171 cross sections for, 344–348 deflection and, 757–765, 790 discontinuity functions for, 757–765, 790 Macaulay functions and, 758920 Index magnitude of resultant force, 161 resultant forces of, 162–165 shear and moment diagram regions, 524–526 shear stress (t), 477 singularity functions, 759–760 stress (s) and, 344–348 torsion and, 477 Dot notation, 27 Dot product, 81–85, 93, 120 Dry friction, 218–235, 237 angles (f) of, 219–221 coefficients of (μ), 219–221, 237 distributed and frictional loads, 218 impending motion (static) and, 219, 221, 222–228, 237 motion (kinetic), 220–221, 237 normal forces and, 218 procedure for analysis of, 225 rigid-body equilibrium and, 218–235, 237 rolling and, 221 slipping (sliding) and, 219–223, 237 theory of, 218 tipping and, 218, 224 Ductile materials, 403–404, 424 E Eccentric loading, 816–821, 825 Effective length, 805 Elastic behavior, 399–401, 431–438, 467 axially loaded members, 431–438, 467 deformation and, 431–438, 467 stress–strain diagrams for, 399–401 stress/strain transformation and, 702–703 Elastic curve, 735–739, 790, 838–839 Electrical-resistance strain gage, 398 Engineering notation, 28 Engineering (nominal) stress/strain, 399 Equilibrium, 174–239, 347–348, 354, 796–797 bifurcation point, 797 column buckling and, 796–797 column buckling and, 796–797 conditions for, 175–176 dry friction and, 218–235 equations of, 187–192, 208–211 free-body diagrams for, 177–185, 203–207 friction force equations and, 218, 223 neutral, 797 procedures for analysis of, 182, 188, 209, 225 rigid bodies, 174–239 scalar equations of, 208 shear stress (t) and, 354 stable, 796 stress (s) and, 347–348, 354 support reactions, 177–179, 203–207, 236–237 three-dimensional rigid-bodies, 203–217, 237 two-dimensional rigid bodies, 176–202, 237 unstable, 219, 796 vector equations of, 208 zero force for, 176 Equivalent system, 138–143 Euler load, 800, 825 Extensometer, 398 External effects of force systems, 138. See also Rotation; Translation F Factor of safety (F.S.), 364–365, 393 Fastener spacing in built-up beams, 597, 605, 723 Fixed supports, 177, 804 Flexibility (force) method, 777–780 Flexural rigidity (EI), 740 Flexure formula, 547–553, 573 bending moment (M) for, 548–549, 573 bending stress from, 547–553, 573 moment of inertia (I) for, 549 neutral axis location for, 548 procedure of analysis for, 550 Force (F), 22–26, 34–95, 119–123, 128–133, 169–170, 176–180, 193–194, 218–237, 243–244, 257–258, 331–332, 379, 392–393, 526. See also Dry friction; Frictional forces; Weight addition of, 38–43 axis, moment of about, 119–123, 170 component vectors of, 38–39 compressive, 243–244, 257–258 concentrated, 23 concept of, 22 concurrent, 50, 61, 93, 193 coplanar, 49–54, 92, 332 couple, moment of, 128–133, 170 deformation from, 379, 393 directed along a line, 73–75, 82, 93 free-body diagrams for, 177–180 frictional, 218–235, 237 gravitational, 25 internal, 180 internal resultant loadings, 331–332, 392 moment (MO) of, 119–123, 170 Newton’s laws and, 24–25 normal (N), 218, 331–332 parallelogram laws for, 36, 40 position vectors and, 70–72, 93 procedure for analysis of, 40 resultants of, 38–39, 50–51, 61, 91–93, 129, 170 rigid-body equilibrium and, 176–180 scalar determination of, 35–36, 119, 128, 169Index 921 Force (continued) shear (V), 331–332 shear and moment diagram regions of concentration, 526 tensile, 243–244, 257–258 triangle rule for, 36–37, 91 two-and three-force members, equilibrium of, 193–194 units of, 26 unknown forces, 177, 188, 203, 206, 209, 236–237, 244 vector determination of, 34–95, 120–121, 128, 169 zero, 176 Force (flexibility) method of analysis, 453–454, 777–780 Force systems, 96–173, 174–239, 242–243, 282 axis, moment of about, 119–123, 170 Cartesian vector formulation, 102–103, 105–107, 169 concurrent, 149, 171, 193, 242, 282 coplanar, 139–140, 149, 161, 177–185, 187–194, 242 couple moments, 128–133, 138–143, 149–154, 170–171, 177 cross product, 101–103, 169 distributed loads, 161–165, 171 dry friction and, 218–235, 243 equilibrium of, 174–239 equivalent, 138–143, 171 external effects from, 138 free-body diagrams for, 177–185, 203–207 frictional forces on, 218–235, 237 moments (MO), 97–107, 119–123, 128–133, 169 parallel, 150, 171, 193 perpendicular, 149 principle of moments, 108–110, 169 principle of transmissibility, 104, 138 procedures for analysis of, 140, 150, 182, 188, 209 resultants, 96–173 rigid-bodies, 174–239 rotational motion and, 138–140, 177, 203 scalar formulation of, 97–100, 119, 128, 169 simplification of, 138–143, 149–154, 171 support reactions and, 177–179, 203–207, 236–237 three-dimensional rigid-bodies, 203–217, 237 translational motion and, 138–140, 177, 203 truss members, 242–243, 282 two-dimensional (coplanar) rigid bodies, 176–202, 236 vector formulation of, 103–105, 120–121, 128, 169 Fracture stress (sf), 401 Frames, 266–281, 283 free-body diagrams for, 266–273, 283 multiforce members of, 266 procedure for analysis of, 269 structural analysis of, 266–281, 283 Free-body diagrams, 177–185, 188, 203–207, 209, 218–219, 236–237, 244–251, 257–262, 266–273, 282–283 center of gravity (G), 180 frames and machines, 266–273, 283 frictional forces, 218–219, 237 idealized models, 180–181 internal forces and, 180 procedures for analysis using, 182, 188, 209 springs, 180 structural analysis using, 244–251, 257–262, 266–273, 282–283 support reactions, 177–179, 203–207 three-dimensional rigid bodies, 203–207, 209, 237 trusses, 244–251, 257–262, 282–283 two-dimensional rigid bodies, 177–185, 188, 236 unknown forces, 177, 188, 203, 206, 209, 236–237 weight (W), 180 Free vector, 128, 138–139 Friction, 218. See also Frictional forces; Dry friction Frictional forces, 218–235, 237. See also Dry friction angles (f) of, 219–221 coefficients of (μ), 219–221, 237 dry friction and, 218–235 equilibrium equations and, 218, 223, 237 free-body diagrams for, 218–219, 237 kinetic (motion), 220–221, 237 normal (N), 218 procedure for analysis of, 225 rigid-body equilibrium and, 218–235, 237 static (impending motion), 219, 221, 222–228 G Gage-length distance (L0), 398 Glulam beams, 722 Gravitational attraction, 25 Gravity. See Center of gravity; Weight Gusset plate, 242 H Hinge supports, 203 Homogeneous material, 346 Hooke’s law, 400, 402, 407–418, 424, 700–701, 713 linear elastic behavior and, 407–418 modulus of elasticity from, 400, 402, 424 shear modulus of elasticity from, 418 stress/strain transformation and, 700–701, 713 Hoop (cylindrical) stress, 610–611, 632 Horizontal angle (u), 60–61 Hydrostatic loading, 703, 713 Hyperbolic functions, 830 I Idealized models of rigid bodies, 180–181 Impending motion, 219, 221, 222–228 Inertia. See Moment of inertia Inflection point, 736–737 In-plane shear strain, 684 In-plane shear stress, 645–651, 711922 Index Integrals, 831 Integration. See Method of integration Internal forces, 180 Internal resultant loadings, 330–343, 392 bending moment (M) in, 331–332 coplanar systems, 332 cross sections for, 330–332, 392 normal force (N) in, 331–332 procedure for analysis of, 333 shear force (V) in, 331–332 three-dimensional components, 331 torsional moment (T) in, 331 Internal torque, 474–475, 492–493, 496 International System (SI) of units, 26–27 Isotropic material, 346 J Joint connections, 242 K Kilogram (kg), unit of, 26 Kinetic frictional forces (motion), 220–221, 237 L Lateral contraction, 416 Lateral deflection, 795–797 Length, units of, 26 Line of action, 35, 119, 138, 149, 162, 169 Linear coefficient of thermal expansion (a), 459 Load-displacement relationships, 444–452, 467, 777–780 axially loaded members, 444–452, 467 beam deflection and, 777–780 statically indeterminate members, 444–452, 467, 777–780 Loads. See Axial loads; Combined loadings; Distributed loadings; Internal resultant loadings Longitudinal axis, 543–545 Longitudinal elongation, 416 Longitudinal shear stress, 577–578 M Macaulay functions, 758 Machines, 266. See also Frames; Structural analysis Magnitude, 35, 38–39, 49, 59, 91–92, 98, 101, 104, 128, 161, 169 arrow notation for, 35, 49 Cartesian vectors, 59, 92 coplanar force systems, 49 couple moments, 128 cross product and, 101 distributed loadings and, 161 moments of a force and, 98, 101, 104 resultant forces, 38–39, 161 scalar determination of, 35, 91, 98, 169 vectors and, 35, 38–39, 49, 59, 91–92, 104 Mass, quantity of, 22 Mass, units of, 26 Material properties, 397–427, 700–707 brittle materials, 405, 424 compression test for, 397–398 dilatation (e), 702–703, 713 ductile materials, 403–404, 424 elastic behavior, 399–401, 702–703 Hooke’s law, 400, 402, 418, 424, 700–701, 713 necking, 401 Poisson’s ratio (ν) for, 416–417, 425 shear stress–strain diagrams for, 418–421, 425 stiffness, 406 strain energy, 407–411, 425 strain hardening, 401, 406 stress and strain transformation effects, 700–707 stress–strain diagrams for, 399–406, 418–421, 424–425 tension test for, 397–398 volume (hydrostatic loading), 703, 713 yielding, 400 Mechanics of materials, 21–25, 328–395. See also Material properties deformation, 379, 393 engineering study of, 21–22, 329 fundamental concepts of, 23–25 internal resultant loadings, 330–343, 392 Newton’s laws for, 24–25 procedure for analysis of problems, 29–30 strain (P), 380–390, 393 stress (s), 344–378, 392–393 Meter (m), unit of, 26 Method of integration, 739–749, 790 boundary conditions, 740 continuity conditions, 740 flexural rigidity (EI), 740 procedure for analysis of, 742 slope equations, 739–749, 790 Method of joints, 244–249, 282 Method of sections, 257–262, 283 Method of superposition. See Superposition Modulus of elasticity (E), 400, 702–703 Modulus of resilience (ur), 407 Modulus of rigidity (G), 418 Modulus of toughness (ut), 408 Mohr’s circle, 661–667, 688–692, 712–713 plane strain, 688–692, 713 plane stress, 661–667, 712 Moment arm (distance), 98 Moment axis (direction), 98 Moment–curvature relationship, 738 Moment diagrams, 735–749, 790 elastic curve, 735–739, 790 inflection point, 736–737Index 923 Moment of inertia (I), 310–314, 319–321, 326, 474, 549, 562–563, 573, 801, 832–833 area (A), 310–314, 319–321, 326, 832–833 column buckling, 801 composite bodies, 319–321, 326 flexure formula and, 549 least, 801 parallel-axis theorem for, 311–314, 326 polar, 310, 326, 475–476 principal axis of, 562–563, 573 procedures for analysis of, 312, 319 product of, 563 torsion formula and, 474 unsymmetric bending and, 562–563, 573 Moments (M), 96–173, 331–332, 526, 580–581. See also Bending moment; Moment of inertia Cartesian vector formulation, 102–103, 105–107, 169 couple, 128–133, 138–143, 149–154, 170–171 cross product for, 101–103 cross-sectional (Q), 580–581 direction of, 98, 101–102, 104, 169 dot product for, 120 force about an axis, 119–123, 170 force and couple systems, simplification of, 138–143, 149–154, 171 internal (M0) loadings, 331–332 magnitude of, 98, 101, 104, 169 principle of, 108–110 principle of transmissibility, 104, 138 procedures for analysis of, 140, 150 resultant (MR), 98–100, 105–107 right-hand rule for, 98, 101–102, 169–170 scalar formulation of, 97–100, 119, 128, 169 shear and moment diagram regions of concentration, 526 sign convention for, 98 torque as, 97 torsional (T ), 331 transverse shear and, 580–581 vector formulation of, 104–107, 120–121, 128, 169 Motion, 24, 138–143, 177, 203, 218–235, 222–228 force and couple system simplification, 138–143 frictional forces and, 218–235, 237 impending, 219, 221, 222–228, 237 kinetic frictional forces, 220–221, 237 Newton’s laws of, 24 rigid-body equilibrium and, 218–235 rolling, 221 rotational, 138–140, 177, 203 slipping (sliding), 219–223 static frictional force and, 219, 221 supports for prevention of, 177, 203 tipping, 218, 224 translational, 138–140, 177, 203 Multiforce members, 266 N Necking, 401 Neutral axis, 543, 548, 562, 565 Neutral equilibrium, 797 Newton (N), unit of, 26 Newton’s law of gravitational attraction, 25 Newton’s laws of motion, 24 Nominal (engineering) stress/strain, 399 Nominal dimensions, 721 Normal force (N), internal resultant loadings, 331–332 Normal strain (P), 380–382, 393 Normal stress (s), 345–352, 392, 642–643 Numerical calculations, engineering use of, 28–29 O Offset method, 403 P Parallel force systems, 150, 193 Parallel-axis theorem, 311–314, 326 Parallelogram law, 36–37, 40, 91 Particles, concept of, 23 Pascal (Pa), unit of, 345 Percent elongation, 403, 424 Percent reduction in area, 403, 424 Perfectly plastic materials, 400 Permanent set, 406 Perpendicular force systems, 149 Pin connections, 242–243 Pin supports, 203, 798–803 Planar trusses, 241 Plane strain, 679–697, 713 absolute maximum shear strain, 696–697, 713 equations for transformation, 680–688 maximum in-plane shear strain, 684 Mohr’s circle for, 688–692, 713 normal and shear strain components, 681–683 principle strains, 684 procedures for analysis of, 688–689 sign convention for, 680 transformation of, 679–688 Plane stress, 637–676, 711–712 absolute maximum shear stress, 673–676, 712 equations for transformation, 642–644 in-plane shear stress, 645–651, 711 Mohr’s circle for, 661–667, 712 normal and shear stress components, 642–643 principle stresses, 645–651 procedures for analysis of, 639, 643, 663–664 sign convention for, 642 transformation of, 637–644924 Index Plastic deformation, 400 Plate girders, 719 Poisson’s ratio (ν), 416–417, 425 Polar moment of inertia, 310, 326, 475–476 Position vectors, 70–72, 93, 169 Power transmission, 482–483, 513 Power-series expansion, 830 Primary beam, 777 Principal axis, 562–563, 573 Principle of moments, 108–110, 169 Principle of transmissibility, 104, 138 Principle strains, 684 Principle stresses, 645–651 Prismatic beams, 720–727 Product of inertia (I), 563 Projection of a moment, 120 Proportional limit (spl), 399, 418 Purlins, 241 Pythagorean theorem, 829 Q Quadratic formula, 830 R Radial stress, 611 Radius of gyration, 801 Rectangular components, 50–51, 58–64, 70–71, 92–93 three dimensional, 58–64, 70–71, 92–93 two dimensional, 50–51, 92 Redundants, 776–779, 791 Relative displacement (d), 431–438 Resultants, 36–39, 50–51, 61–64, 91–93, 96–173, 330–343, 392 axis, moment of force about, 119–123, 161–165 Cartesian vector formulation, 61–62, 93, 102–103, 105– 107, 169 centroid (C) location and, 162 collinear vectors, 37, 91 concurrent forces, 50, 61, 93, 149 coplanar force, 50–51, 92, 149 couple moments, 128–133, 138–143, 149–154, 170–171 cross product, 101–103, 169 direction of, 98, 101–102, 104, 169 distributed loadings, 161–165 equivalent force systems, 138–143 force components, 38–39 force and couple moments,simplification of, 138–143, 149–154, 171 force systems, 96–173 internal loadings, 330–343, 392 magnitude of, 98, 101, 104, 161 moments (MR), 98–100, 105–107, 129–130, 138–143 parallel force systems, 150 parallelogram law for, 36, 40, 91 perpendicular force systems, 149 principle of moments, 108–110 principle of transmissibility, 104, 138 procedures for analysis of, 140, 150 rectangular components, 50–51, 92 right-hand rule for, 98, 101–102, 104, 128 scalar formulation of moment, 98, 128, 169 triangle rule for, 36–37, 91 vector addition for, 36–38, 61–64, 93 vector formulation of moment, 104–107, 128, 169 vector subtraction for, 37 Right-hand rule, 58, 98, 101–102, 104, 128 Rigid bodies, 23, 174–239 center of gravity (G), 180 concept of, 23 dry friction and, 218–235, 237 equations of equilibrium for, 187–192, 208–211 equilibrium of, 174–239 free-body diagrams for, 177–185, 203–207, 236, 237 frictional forces on, 218–235, 237 idealized models of, 180–181 impending motion (static) of, 219, 221, 222–228 internal forces and, 180 procedures for analysis of, 182, 188, 209, 225 rotational motion of, 177, 203 springs, 180 support reactions, 177–179, 203–207, 236–237 three-dimensional, equilibrium of, 203–217, 237 three-force members, 193–194 translational motion of, 177, 203 two-dimensional (coplanar), equilibrium of, 176–202, 236 two-force members, 193–194 weight (W), 180 Rolled shapes, 721 Roller supports, 177 Rolling motion, 221 Roof trusses, 241–242 Rotational motion, 138–140, 177, 203 force and couple system simplification, 138–140 supports for prevention of, 177, 203 Rounding off numbers, 29 S Saint-Venant’s principle, 429–431, 466 Scalar triple product, 120 Scalars, 35–36, 49, 91, 97–100, 119, 128, 169, 208 coplanar forces, notation for, 49 couple moments, 128 division of a vector by, 36 equations of equilibrium, 208 moment of force about an axis, 119 moment of a force, formulation of, 97–100, 119, 128, 169 multiplication of a vector by, 36 quantity, 35, 91 sign convention for, 98, 169Index 925 Secant formula, 816–821, 825 Seconds (s), unit of, 26 Section modulus (S), 720 Shafts, 471–473, 482–483, 492–499, 506–509, 513 angle of twist, 472–473, 492–499, 513 circular, 471–473, 513 internal torque, 492–493, 496 power transmission, design for, 482–483 procedures for analysis of, 496, 507 statically indeterminate, 506–509, 513 torsional deformation, 471–473, 513 Shear and moment diagrams, 517–533, 572 beams, 517–533, 572 concentrated forces and moments, 526 distributed loads, 524–526 graphical methods for, 524–533, 572 procedures for analysis of, 519, 527 sign convention for, 518, 526, 572 Shear flow (q), 596–600, 605 Shear force (V), internal resultant loadings, 331–332 Shear formula, 578–589, 605 Shear modulus (G), 418, 425, 702, 713 Shear strain (g), 381, 393, 474, 679–687, 696–697, 713 absolute maximum, 696–697, 713 deformation and, 381, 393 linear variation in, 474 maximum in-plane, 684 plane strain components, 681–683 plane strain transformation and, 679–687, 696–697, 713 Shear stress (t), 345, 353–357, 392, 474, 477, 577–579, 642–651, 673–676, 711–712. See also Transverse shear absolute maximum, 673–676, 712 beams 577–579 direct (simple), 353 distribution of, 345, 477, 578–579 equilibrium, 354 in-plane, 645–651, 711 linear variation in, 474 longitudinal, 577–578 plane stress components, 642–643 plane stress transformation and, 642–651, 673–676, 711–712 procedure for analysis of, 355 torsion and, 474, 477 Shear stress–strain diagrams, 418–421, 425 Significant figures, 28–29 Sine law, 40 Singularity functions, 759–760 Slenderness ratio, 801–802 Sliding vector, 104, 138 Slipping (sliding), 219–223, 237 Slope equations, 739–749, 790, 838–839 Small strain analysis, 382 Specific weight (constant density), 301 Spherical vessels, 611, 632 Springs, free-body diagrams of, 180 Stable equilibrium, 796 Static frictional forces (impending motion), 219, 221, 222–228, 237 Statically indeterminate members, 446–454, 467, 506–509, 513, 776–785, 791 axially loaded, 446–454, 467 beams, 776–785, 791 compatibility (kinematic) conditions for, 447–452, 467 deflection of, 776–785, 791 degree of indeterminacy, 776 force (flexibility) method of analysis, 453–454, 777–780 load-displacement relationships, 447–452, 467, 777–780 method of superposition for, 446–452, 776–785, 791 procedures for analysis of, 448, 507, 780 redundants (reactions) of, 776–779, 791 shafts, 506–509, 513 torque loaded, 506–509, 513 Steel, stress–strain diagram for, 402 Steel sections, structural design and, 721 Stiffness, 406 Straight members. See Beams Strain (P), 380–390, 393, 399, 684. See also Plane strain Cartesian components, 381 deformation and, 380–385, 393 dimensionless quantity of, 380, 393 nominal (engineering), 399 normal (P), 380–382, 393 principle, 684 shear (g), 381, 393 small strain analysis, 382 units of, 380 Strain energy, 407–411, 425 Strain hardening, 401, 406 Strain rosettes, 698–699 Stress (s), 344–378, 392–393, 399–400, 459–462, 467, 610–611, 616–623, 632, 645–651, 700, 723, 733. See also Plane stress allowable (sallow), 364–365, 723, 733 allowable stress design (ASD), 364–371, 393 axially loaded bars, 346–352, 392 bending, 723, 733 biaxial, 611 combined loadings and, 610–611, 616–623, 632 cylindrical (hoop), 610–611, 632 equilibrium and, 347–348, 354 factor of safety (F.S.), 364–365, 393 loading distribution and, 344–345 nominal (engineering), 399 normal (s), 345–352, 392 principle, 645–651 procedure for analysis of, 349, 355, 366 radial, 611926 Index shear (t), 345, 353–357, 392, 723, 733 state of, 345, 616–623, 632 thermal, 459–462, 467 triaxial, 700 ultimate (su), 400 uniaxial, 348 units of, 345 Stress and strain transformation, 636–715 absolute maximum shear strain, 696–697, 713 absolute maximum shear stress, 673–676, 712 bulk modulus (k) and, 703, 713 dilatation (e), 702–703, 713 equations for, 642–644, 680–688 Hooke’s law and, 700–701, 713 in-plane shear stress, 645–651, 711 material property relationships, 700–707 modulus of elasticity (E) and, 702–703 Mohr’s circle for, 661–667, 688–692, 712–713 plane strain, 679–697, 713 plane stress, 637–676, 711–712 principle stresses, 645–651 procedures for analysis of, 639, 643, 663–664, 688–689 shear modulus (G) and, 702, 713 strain rosettes, 698–699 triaxial stress, 700 Stress–strain diagrams, 399–421, 424–425 brittle material behavior from, 405, 424 conventional, 399–406, 424 ductile material behavior from, 403–404, 424 elastic behavior, 399–401 fracture stress (sf), 401 modulus of resilience (ur), 407 modulus of rigidity (G), 418 modulus of toughness (ut), 408 nominal (engineering) stress/strain for, 399 proportional limit (spl), 399, 418 shear, 418–421, 425 steel, 402 true, 401 ultimate stress (su), 401, 418 yield point (sY), 400, 406, 408, 424–425 Young’s modulus of elasticity (E), 400 Stress trajectories, 719 Structural analysis, 240–285 frames, 266–281, 283 free-body diagrams for, 244–251, 257–262, 266–273, 282–283 machines, 266–281, 283 method of joints, 244–249, 282 method of sections, 257–262, 283 procedures for, 245, 259, 269 trusses, 241–265, 282–283 zero-force members, 250–252 Structural design, 242–243, 364–371, 393, 482–483, 716–733 allowable bending and shear stress, 723, 733 allowable stress design (ASD), 364–371, 393 beams, 716–733 prismatic beams, 720–727 section modulus (S), 720 shafts, 482–483 stress trajectories, 719 trusses, 242–243 Superposition, 446–452, 467, 768–772, 776–785, 791 axial loads and, 446–452, 467 beams, 768–772, 776–785, 791 deflection and, 768–772, 776–785, 791 displacement (d) and, 446, 467 force (flexibility) method for, 777–780 load-displacement relationships, 446–452, 467, 777–780 method of, 768–772, 776–785, 791 primary beam for, 777 principle of, 446, 467 procedure for analysis of, 780 redundants (reactions) from, 776–779, 791 statically indeterminate members, 447–452, 467, 776–785, 791 Support reactions, 177–179, 203–207, 236–237 free-body diagrams for, 177–179, 203–207 prevention of motion by, 177, 203 three-dimensional rigid-bodies, 203–207, 237 two-dimensional rigid bodies, 177–179, 236 types of, 178–179, 204–205 T Tensile forces, 243–244, 257–258 Tension test, 397–398 Thermal stress, 459–462, 467 Thin-walled pressure vessels, 609–612, 632 Three-dimensional rigid-bodies, 203–217, 237 equations of equilibrium, 208–211, 237 equilibrium of, 203–217 free-body diagrams for, 203–207 procedure for analysis of, 209 support reactions, 203–207, 237 unknown forces in, 203, 206, 209, 237 Time, units of, 26 Tipping, 218, 224 Torque, 97, 471. See also Moments (M) Torsion, 470–515 angle of twist, 472–473, 492–499, 513 circular shafts, 471–473, 513 constant cross-sectional area (A), 493–494 deformation, 471–473, 513 internal torque and, 474–475, 492–493, 496 polar moment of inertia and, 310, 326, 475–476 power transmission, 482–483, 513Index 927 Torsion (continued) procedures for analysis of, 478, 496, 507 shafts, 471–473, 482–483, 506–509, 513 shear stress distribution, 477 statically indeterminate members, 506–509, 513 torsion formula, 474–481, 513 Torsional moment (T), internal resultant loadings, 331 Translational motion, 138–140, 159, 177, 203 force and couple system simplification, 138–140 supports for prevention of, 177, 203 Transmissibility, principle of, 104, 138 Transverse shear, 576–607 beams, 576–607 built-up members, 596–600, 605 cross-sectional moment (Q), 580–581 longitudinal shear stress and, 577–578 procedures for analysis of, 584 shear flow (q), 596–600, 605 shear formula for, 578–589, 605 shear-stress distribution, 578–579, 605 Triangle rule, 36–37, 91 Triaxial stress, 700 Trigonometric functions and identities, 829–830 True stress–strain diagram, 401 Trusses, 241–265, 282–283 compressive forces, 243–244, 257–258 design assumptions, 242–243 forces determined by inspection, 244, 250, 258 free-body diagrams for, 244–251, 257–262, 282–283 joint connections, 242 method of joints, 244–249, 282 method of sections, 257–262, 283 procedures for analysis of, 245, 259 simple, 241–243, 282 structural analysis of, 241–265, 282–283 tensile forces, 243–244, 257–258 zero-force members, 250–252 Twisting, 471–473 Two-dimensional (coplanar) rigid bodies, 176–202, 236 equations of equilibrium, 187–192 equilibrium of, 176–202, 236 free-body diagrams for, 177–185 procedures for analysis of, 182, 188 support reactions, 177–179, 236 three-force members, 193–194 two-force members, 193–194 unknown forces in, 177, 188, 236 U Ultimate stress (su), 401, 418 Uniaxial stress, 348 Uniform deformation, 346 Unit vectors, 50, 58–61, 73, 73–75, 92 Units of measurement, 26–27, 28, 345, 380, 393, 482, 721 base units, 26 derived units, 26 dimensional homogeneity, 28 dimensionless quantity and, 380, 393 foot-pound-second system, 345 force, 26 International System (SI), 26–27 length, 26 mass, 26 nominal dimensions, 721 power, 482 rules for use, 27 SI prefixes, 26–27 strain, 380 stress, 345 time, 26 weight, 26 Unstable equilibrium, 219, 796 Unsymmetric bending, 562–568, 573 bending moment (M) of, 562–568, 573 moment arbitrarily applied, 564 neutral axis orientation, 565 principal axis, moment applied to, 562–563, 573 product of inertia (I ) for, 563 V Varignon’s theorem, 108–110 Vectors, 34–95, 101–107, 120–121, 128, 138–143, 169–170, 208 addition, 36–37, 38–43, 49–54, 61–64, 91–93 angle (u), 35, 59–61, 82, 92–93 arrow notation, 35, 49, 91 Cartesian, 50, 58–64, 70–75, 81, 92–93, 102–103, 105–107, 169 collinear, 37, 91 component, 36, 38–39, 49–54 coplanar forces, 49–54 couple moments, 128, 170 cross product for, 101–103, 169–170 directed along a line, 73–75, 82, 93 direction of, 35, 38, 49, 58–61, 70–71, 73, 101–102, 104 division of by a scalar, 36, 91 dot product, 81–85, 93, 120 equations of equilibrium, 208 force and, 34–95, 128, 169 force and couple systems, 138–143 free, 128, 138–139 line of action, 35, 119, 138, 169 magnitude of, 35, 49, 59, 91–92, 101, 104 moment of force about an axis, 120–121, 169 multiplication by a scalar, 36, 91 parallelogram law for, 36–37, 40, 91 position, 70–72, 93, 169 procedures of analysis for, 40928 Index rectangular components, 50–51, 58–64, 70–71, 92–93 resultant moments and, 105–107, 169 resultants of a force, 36–38, 50–51, 91, 93, 101–107, 129– 130, 169 right-hand rule for, 58 scalars and, 35–36, 49, 91 several forces, 39 sliding, 104, 138 subtraction, 37 three-dimensional components, 58–64, 70–71, 92–93 triangle rule for, 36–37, 91 two-dimensional components, 50–51, 92 unit, 50, 58–61, 73–75, 92 x, y, z coordinates, 58–60, 70, 92–93 Vertical angle (f), 60–61 Volume (V), centroid of, 289, 325, 832–833 Volume changes (hydrostatic loading), 703, 713 W Warping, 578 Weight (W), 25, 26, 180, 287–289, 325 center of gravity (G) and, 180, 287–289, 325 free-body diagrams and, 180 gravitational force and, 25 rigid-body equilibrium and, 180 units of, 26 Wide-flange sections, 834–837 Wood, ductility of, 404 Wood sections, structural design and, 721 X x, y, z coordinates, 58–60, 70, 92–93 Y Yield point (sY), 400, 406, 408, 424–425 Yield strength, 403–404 Yielding, 400 Young’s modulus (E), 400 Z Zero force, 176. See also Equilibrium Zero-force members, 250–252
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